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Mirrors > Home > MPE Home > Th. List > Mathboxes > mvrsval | Structured version Visualization version GIF version |
Description: The set of variables in an expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mvrsval.v | ⊢ 𝑉 = (mVR‘𝑇) |
mvrsval.e | ⊢ 𝐸 = (mEx‘𝑇) |
mvrsval.w | ⊢ 𝑊 = (mVars‘𝑇) |
Ref | Expression |
---|---|
mvrsval | ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2nd ‘𝑋) ∩ 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mvrsval.w | . . 3 ⊢ 𝑊 = (mVars‘𝑇) | |
2 | elfvex 6131 | . . . . 5 ⊢ (𝑋 ∈ (mEx‘𝑇) → 𝑇 ∈ V) | |
3 | mvrsval.e | . . . . 5 ⊢ 𝐸 = (mEx‘𝑇) | |
4 | 2, 3 | eleq2s 2706 | . . . 4 ⊢ (𝑋 ∈ 𝐸 → 𝑇 ∈ V) |
5 | fveq2 6103 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇)) | |
6 | 5, 3 | syl6eqr 2662 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸) |
7 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇)) | |
8 | mvrsval.v | . . . . . . . 8 ⊢ 𝑉 = (mVR‘𝑇) | |
9 | 7, 8 | syl6eqr 2662 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉) |
10 | 9 | ineq2d 3776 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (ran (2nd ‘𝑒) ∩ (mVR‘𝑡)) = (ran (2nd ‘𝑒) ∩ 𝑉)) |
11 | 6, 10 | mpteq12dv 4663 | . . . . 5 ⊢ (𝑡 = 𝑇 → (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡))) = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉))) |
12 | df-mvrs 30640 | . . . . 5 ⊢ mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡)))) | |
13 | fvex 6113 | . . . . . . 7 ⊢ (mEx‘𝑇) ∈ V | |
14 | 3, 13 | eqeltri 2684 | . . . . . 6 ⊢ 𝐸 ∈ V |
15 | 14 | mptex 6390 | . . . . 5 ⊢ (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉)) ∈ V |
16 | 11, 12, 15 | fvmpt 6191 | . . . 4 ⊢ (𝑇 ∈ V → (mVars‘𝑇) = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉))) |
17 | 4, 16 | syl 17 | . . 3 ⊢ (𝑋 ∈ 𝐸 → (mVars‘𝑇) = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉))) |
18 | 1, 17 | syl5eq 2656 | . 2 ⊢ (𝑋 ∈ 𝐸 → 𝑊 = (𝑒 ∈ 𝐸 ↦ (ran (2nd ‘𝑒) ∩ 𝑉))) |
19 | fveq2 6103 | . . . . 5 ⊢ (𝑒 = 𝑋 → (2nd ‘𝑒) = (2nd ‘𝑋)) | |
20 | 19 | rneqd 5274 | . . . 4 ⊢ (𝑒 = 𝑋 → ran (2nd ‘𝑒) = ran (2nd ‘𝑋)) |
21 | 20 | ineq1d 3775 | . . 3 ⊢ (𝑒 = 𝑋 → (ran (2nd ‘𝑒) ∩ 𝑉) = (ran (2nd ‘𝑋) ∩ 𝑉)) |
22 | 21 | adantl 481 | . 2 ⊢ ((𝑋 ∈ 𝐸 ∧ 𝑒 = 𝑋) → (ran (2nd ‘𝑒) ∩ 𝑉) = (ran (2nd ‘𝑋) ∩ 𝑉)) |
23 | id 22 | . 2 ⊢ (𝑋 ∈ 𝐸 → 𝑋 ∈ 𝐸) | |
24 | fvex 6113 | . . . . 5 ⊢ (2nd ‘𝑋) ∈ V | |
25 | 24 | rnex 6992 | . . . 4 ⊢ ran (2nd ‘𝑋) ∈ V |
26 | 25 | inex1 4727 | . . 3 ⊢ (ran (2nd ‘𝑋) ∩ 𝑉) ∈ V |
27 | 26 | a1i 11 | . 2 ⊢ (𝑋 ∈ 𝐸 → (ran (2nd ‘𝑋) ∩ 𝑉) ∈ V) |
28 | 18, 22, 23, 27 | fvmptd 6197 | 1 ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2nd ‘𝑋) ∩ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 ↦ cmpt 4643 ran crn 5039 ‘cfv 5804 2nd c2nd 7058 mVRcmvar 30612 mExcmex 30618 mVarscmvrs 30620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-mvrs 30640 |
This theorem is referenced by: mvrsfpw 30657 msubvrs 30711 |
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